collect

Collect the coefficients of identical powers of the default variable in a symbolic expression. Here, collect returns the expression as a polynomial in terms of x by collecting the coefficients of x^2 and x.

syms x
P = (exp(x) + x)*(x + 2)

P = $$\left(x+{\mathrm{e}}^x\right)\hspace{0.17em}\left(x+2\right)$$

C = collect(P)

C = $$x^2+\left({\mathrm{e}}^x+2\right)\hspace{0.17em}x+2\hspace{0.17em}{\mathrm{e}}^x$$

Because you did not specify the variable, collect uses the default variable determined by symvar. For this expression, the default variable is x.

defaultvar = symvar(P)

defaultvar = $$x$$

Collect the coefficients of identical powers of a specified variable in a symbolic expression by specifying the variable as the second argument to collect.

Create a symbolic expression in terms of the variables x and y.

syms x y
P = x^2*y + y*x - x^2 - 2*x

P = $$x\hspace{0.17em}y-2\hspace{0.17em}x+x^2\hspace{0.17em}y-x^2$$

Collect the coefficients of identical powers of x. Here, collect returns the expression as a polynomial in terms of x by collecting the coefficients of x^2 and x.

Cx = collect(P,x)

Cx = $$\left(y-1\right)\hspace{0.17em}{x}^2+\left(y-2\right)\hspace{0.17em}x$$

Collect the coefficients of identical powers of y. Here, collect returns the expression as a polynomial in terms of y by collecting the coefficients of y.

Cy = collect(P,y)

Cy = $$\left(x^2+x\right)\hspace{0.17em}y-x^2-2\hspace{0.17em}x$$

You can also collect coefficients in terms of multiple variables by specifying the second argument as a vector of variables. Create another symbolic expression in terms of the variables x and y. Then collect the coefficients of identical powers of x and y. Here, collect returns the expression as a polynomial in terms of two variables, x and y, by collecting the coefficients of x^2 and x*y.

syms a b
Q = a^2*x*y + a*b*x^2 + a*x*y + x^2

Q = $$y\hspace{0.17em}{a}^2\hspace{0.17em}x+b\hspace{0.17em}a\hspace{0.17em}{x}^2+y\hspace{0.17em}a\hspace{0.17em}x+x^2$$

Cxy = collect(Q,[x y])

Cxy = $$\left(a\hspace{0.17em}b+1\right)\hspace{0.17em}{x}^2+\left(a^2+a\right)\hspace{0.17em}x\hspace{0.17em}y$$

Collect the coefficients of a symbolic expression in terms of i, and then in terms of pi.

syms x y
P = y*pi*(pi - 1i) + x*(pi + 1i) + 3*pi

P = $$3\hspace{0.17em}\pi +x\hspace{0.17em}\left(\pi +\mathrm{i}\right)+\pi \hspace{0.17em}y\hspace{0.17em}\left(\pi -\mathrm{i}\right)$$

Ci = collect(P,1i)

Ci = $$\left(x-\pi \hspace{0.17em}y\right)\hspace{0.17em}\mathrm{i}+3\hspace{0.17em}\pi +\pi \hspace{0.17em}x+y\hspace{0.17em}{\pi }^2$$

Cpi = collect(P,pi)

Cpi = $$y\hspace{0.17em}{\pi }^2+\left(x+3-y\hspace{0.17em}\mathrm{i}\right)\hspace{0.17em}\pi +x\hspace{0.17em}\mathrm{i}$$

You can specify the symbolic expression or function in terms of which you collect coefficients as the second argument of collect.

Expand the expression sin(x + 3*y) by using expand. Then collect the coefficients in terms of sin(x).

syms x y
P = expand(sin(x + 3*y))

P = $$4\hspace{0.17em}\sin \left(x\right)\hspace{0.17em}{\cos \left(y\right)}^3+4\hspace{0.17em}\cos \left(x\right)\hspace{0.17em}\sin \left(y\right)\hspace{0.17em}{\cos \left(y\right)}^2-3\hspace{0.17em}\sin \left(x\right)\hspace{0.17em}\cos \left(y\right)-\cos \left(x\right)\hspace{0.17em}\sin \left(y\right)$$

C = collect(P,sin(x))

C = $$\left(4\hspace{0.17em}{\cos \left(y\right)}^3-3\hspace{0.17em}\cos \left(y\right)\right)\hspace{0.17em}\sin \left(x\right)+4\hspace{0.17em}\cos \left(x\right)\hspace{0.17em}{\cos \left(y\right)}^2\hspace{0.17em}\sin \left(y\right)-\cos \left(x\right)\hspace{0.17em}\sin \left(y\right)$$

Collect the coefficients in terms of both sin(x) and sin(y) by specifying a vector input.

Cxy = collect(P,[sin(x) sin(y)])

Cxy = $$\left(4\hspace{0.17em}{\cos \left(y\right)}^3-3\hspace{0.17em}\cos \left(y\right)\right)\hspace{0.17em}\sin \left(x\right)+\left(4\hspace{0.17em}\cos \left(x\right)\hspace{0.17em}{\cos \left(y\right)}^2-\cos \left(x\right)\right)\hspace{0.17em}\sin \left(y\right)$$

You can also collect coefficients in terms of the symbolic function y(x) in a symbolic expression.

syms y(x)
P = y^2*x + y*x^2 + y*sin(x) + x*y

P(x) = $$x\hspace{0.17em}{y\left(x\right)}^2+x^2\hspace{0.17em}y\left(x\right)+\sin \left(x\right)\hspace{0.17em}y\left(x\right)+x\hspace{0.17em}y\left(x\right)$$

Cy = collect(P,y)

Cy(x) = $$x\hspace{0.17em}{y\left(x\right)}^2+\left(x+\sin \left(x\right)+x^2\right)\hspace{0.17em}y\left(x\right)$$

If you specify a matrix of symbolic inputs, collect acts element-wise on the matrix.

syms x y
P = [(x + 1)*(y + 1), x^2 + x*(x -y);
     2*x*y - x, x*y + x/y]

P = 
$$\left(\begin{array}{cc}\left(x+1\right)\hspace{0.17em}\left(y+1\right)& x\hspace{0.17em}\left(x-y\right)+x^2\\ 2\hspace{0.17em}x\hspace{0.17em}y-x& x\hspace{0.17em}y+\frac{x}{y}\end{array}\right)$$

C = collect(P,x)

C = 
$$\left(\begin{array}{cc}\left(y+1\right)\hspace{0.17em}x+y+1& 2\hspace{0.17em}{x}^2+\left(-y\right)\hspace{0.17em}x\\ \left(2\hspace{0.17em}y-1\right)\hspace{0.17em}x& \left(y+\frac{1}{y}\right)\hspace{0.17em}x\end{array}\right)$$

Collect the coefficients in terms of calls to a particular function by specifying the function name as a string scalar in the second argument. Collect the coefficients of function calls with respect to multiple functions by specifying the multiple functions as a string array.

Collect the coefficients in terms of calls to the sin function in P, where P is a symbolic expression that contains multiple calls to different functions.

syms a b c d e f x
P = a*sin(2*x) + b*sin(2*x) + c*cos(x) + ...
    d*cos(x) + e*sin(3*x) + f*sin(3*x)

P = $$a\hspace{0.17em}\sin \left(2\hspace{0.17em}x\right)+b\hspace{0.17em}\sin \left(2\hspace{0.17em}x\right)+e\hspace{0.17em}\sin \left(3\hspace{0.17em}x\right)+f\hspace{0.17em}\sin \left(3\hspace{0.17em}x\right)+c\hspace{0.17em}\cos \left(x\right)+d\hspace{0.17em}\cos \left(x\right)$$

C1 = collect(P,"sin")

C1 = $$\left(a+b\right)\hspace{0.17em}\sin \left(2\hspace{0.17em}x\right)+\left(e+f\right)\hspace{0.17em}\sin \left(3\hspace{0.17em}x\right)+c\hspace{0.17em}\cos \left(x\right)+d\hspace{0.17em}\cos \left(x\right)$$

Collect the coefficients in terms of calls to both the sin and cos functions in P.

C2 = collect(P,["sin" "cos"])

C2 = $$\left(a+b\right)\hspace{0.17em}\sin \left(2\hspace{0.17em}x\right)+\left(e+f\right)\hspace{0.17em}\sin \left(3\hspace{0.17em}x\right)+\left(c+d\right)\hspace{0.17em}\cos \left(x\right)$$

Create a symbolic expression in terms of x and y.

syms x y
P = -12/((x - 2)*(x + 2)) + 8/(y + 2)

P = 
$$\frac{8}{y+2}-\frac{12}{\left(x-2\right)\hspace{0.17em}\left(x+2\right)}$$

Collect the coefficients of identical powers of x and then y in the symbolic expression P. Here, the collect function returns a rational function in the form of the division of two polynomials. It collects the coefficients of identical terms with positive integer powers in the numerator and denominator separately.

Cx = collect(P,x)

Cx = 
$$\frac{8\hspace{0.17em}{x}^2-12\hspace{0.17em}y-56}{\left(y+2\right)\hspace{0.17em}{x}^2-4\hspace{0.17em}y-8}$$

Cy = collect(P,y)

Cy = 
$$\frac{-12\hspace{0.17em}y+8\hspace{0.17em}{x}^2-56}{\left(x^2-4\right)\hspace{0.17em}y+2\hspace{0.17em}{x}^2-8}$$

If the expression cannot be expressed as a rational function that is the division of two polynomials with positive integer powers on the indeterminates, then collect might not collect the coefficients of identical powers.

For example, create a symbolic expression that involves the square root of x. Here, collect does not collect identical powers of x.

Q = -12/(sqrt(x)*(x + 2)) + 8/(y + 2)

Q = 
$$\frac{8}{y+2}-\frac{12}{\sqrt{x}\hspace{0.17em}\left(x+2\right)}$$

C = collect(Q,x)

C = 
$$\frac{16\hspace{0.17em}\sqrt{x}-12\hspace{0.17em}y+8\hspace{0.17em}{x}^{3/2}-24}{2\hspace{0.17em}\sqrt{x}\hspace{0.17em}y+x^{3/2}\hspace{0.17em}y+4\hspace{0.17em}\sqrt{x}+2\hspace{0.17em}{x}^{3/2}}$$